12. Quantum control

In this section, we introduce the theoretical background required to understand the quantum-control features of QTCAD. In particular, this theoretical introduction will help understand the electric-dipole spin resonance (EDSR) tutorials: Electric dipole spin resonance—Dynamics, Electric dipole spin resonance—Noise, Electric dipole spin resonance and Rabi oscillations, and Charge noise in quantum dots.

In EDSR, a time-dependent voltage bias is used to manipulate the spin degree of freedom of an electron or hole. Electrical spin control is made possible by spin–orbit coupling—this spin–orbit coupling may arise from specific material properties or be generated artificially by a micromagnet. Here, we concentrate on the latter situation, because micromagnet EDSR has been demonstrated in a wide range of spin-qubit designs and materials.

As seen in the above-mentioned tutorials, QTCAD simulations of EDSR are run via the Perturbation module. The magnetic field can be accounted for using a Zeeman perturbation (see Electric dipole spin resonance and Rabi oscillations) or by including it directly into the Schrödinger equation via the set_Bfield device method (see Electric dipole spin resonance—Dynamics). The drive is accounted for through a Gate perturbation.


It is also possible to simulate EDSR mediated by inherent material spin–orbit coupling using QTCAD. Spin–orbit coupling can be included in the device using the set_soc device method. Simulating EDSR in this case could be achieved by following the Electric dipole spin resonance—Dynamics tutorial and replacing the section where the magnetic field is set with one where spin–orbit coupling is set. The drive would be simulated in the same way, i.e. using a Gate perturbation.

The Zeeman Hamiltonian and the logical spin qubit

Previous sections have focused mostly on the charge degree of freedom. In contrast, here, we consider a qubit encoded in the spin degree of freedom of a single electron.

The Zeeman Hamiltonian for a single electron in a spatially-varying magnetic field \(\mathbf B(\mathbf r)\) is

(12.1)\[H_{\textrm Z} = \frac12\mu_B\sum_{ij}\sum_{ss'\in\{\uparrow,\downarrow\}} \int d\mathbf r\; F_i^\ast(\mathbf r)F_j(\mathbf r)\; \mathbf B(\mathbf r) \cdot\mathbf g\cdot \boldsymbol\sigma\; |is\rangle\langle js'|,\]

where \(\mu_B\) is the Bohr magneton, \(F_i(\mathbf r)\) is the \(i\)-th eigenfunction of the single-particle time-independent effective Schrödinger equation for the orbital degree of freedom (i.e. without spin, see Schrödinger solvers), \(\mathbf g\) is the effective Landé \(g\)-tensor, and \(\boldsymbol \sigma\equiv (\sigma_x, \sigma_y, \sigma_z)\) is the vector of Pauli matrices

(12.2)\[\sigma_x = |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow|, \qquad \sigma_y = -i|\uparrow\rangle\langle\downarrow| + i|\downarrow\rangle\langle\uparrow|, \qquad \sigma_z = |\uparrow\rangle\langle\uparrow| - |\downarrow\rangle\langle\downarrow|,\]

with \(|\uparrow\rangle\) (\(|\downarrow\rangle\)) being the \(+1\) (\(-1\)) eigenstate of \(\sigma_z\).

In Eq. (12.1), \(i\) and \(j\) run over all eigenfunctions of the single-electron orbital Hamiltonian. In practice, in QTCAD, we truncate this sum to a finite number of orbitals to treat the problem numerically.

From Eq. (12.1), we see that a spatially varying magnetic field may couple the spin and orbital degrees of freedom; for example, this Hamiltonian contains terms causing transitions between orbital states accompanied by a spin-flip. This artificial spin–orbit coupling is, in fact, a useful resource for quantum control of spin qubits, in particular to implement EDSR [PLadriereOT+08, TvdWOT06].

When calling the solve method of a MicromagnetEDSR object, the first computational steps executed by QTCAD are to assemble the Zeeman Hamiltonian defined in Eq. (12.1) in the truncated single-electron orbital eigenstate basis and diagonalize it. Here, we label the resulting eigenstates as \(|k\rangle\), with \(k\;\in\;\{0,1,2,...\}\). Typically, a Loss–DiVincenzo spin qubit is encoded in the ground and first excited state of this basis. Because the energy-level splittings arising from the orbital degree of freedom are typically orders of magnitude larger than the Zeeman splittings (\(\textrm{meV}\) instead of \(\mu \textrm{eV}\)), these first two levels are almost purely of a spin nature, even though a small orbital component may remain, which may be exploited for quantum control by purely electrical means, as will be seen in the next subsection.

Control through gate-bias modulations

Here, we describe how quantum control due to gate-bias modulations may be modelled theoretically, while accounting for realistic device geometry.

Time-dependent Hamiltonian

We consider an initial static gate bias configuration defined by the vector

\[\boldsymbol \varphi^\textrm{bias}_0= [\varphi^\mathrm{bias}_1, \varphi^\mathrm{bias}_2,...,\varphi^\mathrm{bias}_{N_\textrm{gates}}]^T,\]

where \(\varphi^\mathrm{bias}_G\) is the applied potential on gate \(G\;\in\;\{1,2,...,N_\mathrm{gates}\}\), with \(N_\mathrm{gates}\) being the total number of gates in the device.

In this static gate bias configuration, the potential \(\varphi(\mathbf r)\equiv \varphi_0(\mathbf r)\) is obtained by solving the non-linear Poisson equation (see Poisson solvers). The corresponding confinement potential energy is \(V_\mathrm{conf}(\mathbf r)\equiv V_0(\mathbf r)\). The dependence of \(V_0(\mathbf r)\) on \(\varphi_0(\mathbf r)\) varies according to the model used for band alignment across heterojunctions; see Band alignment in heterostructures.

We then consider a time-dependent modulation of the gate biases described by

\[\delta\boldsymbol\varphi^\textrm{bias}(t) \equiv \boldsymbol \varphi^\textrm{bias}(t) -\boldsymbol\varphi^\textrm{bias}_0(t).\]

For example, one of the applied gate potentials may be modulated sinusoidally with respect to its initial static value. The potential corresponding to the solution of Poisson’s equation at each time is written as \(\varphi_\textrm{mod}(\mathbf r,t)\), and the corresponding confinement potential is \(V_\textrm{mod}(\mathbf r,t)\). We may then write a time-dependent perturbation to the Hamiltonian corresponding to the static gate bias configuration as

(12.3)\[\delta \hat V(t) \equiv \hat V_\textrm{mod}(t)-\hat V_0,\]

where \(\hat V_0\) and \(\hat V_\textrm{mod}(t)\) are the quantum operators corresponding to the confinement potentials \(V_0(\mathbf r)\) and \(V_\textrm{mod}(\mathbf r,t)\), respectively.

The perturbation to the confinement potential may be expanded over the truncated basis set formed by the eigensolutions of the Zeeman Hamiltonian, Eq. (12.1), resulting in

(12.4)\[\delta \hat V(t) = \sum_{k\ell}\int d\mathbf r\; F_k^\ast(\mathbf r) \left[V_\textrm{mod}(\mathbf r,t)-V_0(\mathbf r)\right] F_\ell(\mathbf r) |k\rangle\langle\ell|,\]

where \(|k\rangle\) and \(|\ell\rangle\) are the eigenstates of the Zeeman Hamiltonian.

Time-dependent Schrödinger’s equation

The time evolution of the quantum state of the system in the Hilbert space defined by the truncated basis set \(\{|k\rangle\}\) is given by the solution of the time-dependent Schrödinger equation

(12.5)\[i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle= \hat H(t)|\psi(t)\rangle,\]

under the initial condition \(|\psi(0)\rangle\equiv |\psi_0\rangle\) and where we have introduced the time-dependent Hamiltonian

(12.6)\[\hat H (t)= \hat H_0 + \delta \hat V(t),\]

with \(\hat H_0\) the Zeeman Hamiltonian \(H_Z\) (Eq. (12.1)) in its (diagonal) eigenbasis.

From these expressions, a numerical solution of the time-dependent Schrödinger equation may be used to describe EDSR in realistic device geometries and confinement potential configurations arising from user-specified gate biases, including arbitrary spatially dependent magnetic fields.

Relevant measures of performance

The performance of an EDSR implementation may be quantified using several figures of merit.

From elementary quantum mechanics, the expectation value of an operator \(\hat O\) is given by

(12.7)\[\langle \hat O\rangle = \langle\psi(t)|\hat O|\psi(t)\rangle,\]

while the probability \(P_k(t)\) of measuring the system in its eigenstate \(|k\rangle\) is given by

(12.8)\[P_k(t) = |\langle k|\psi(t)\rangle|^2.\]

For EDSR, it is useful to calculate the expectation value of the Pauli operators defined in the basis of the qubit logical states, typically \(|k=0\rangle\) and \(|k=1\rangle\). It may also be useful to calculate \(P_k(t)\) with \(k\) outside the qubit logical subspace to estimate the magnitude of leakage outside the computational subspace under strong EDSR driving.

Another useful quantity is the fidelity of a quantum-logic gate. For pure quantum states \(\varrho = |\psi_\varrho\rangle\langle\psi_\varrho|\) and \(\chi = |\psi_\chi\rangle\langle\psi_\chi|\), the fidelity is

(12.9)\[F(\varrho,\chi) \equiv |\langle \psi_\varrho|\psi_\chi\rangle|^2.\]

In particular, let us consider the case of a target quantum-logic gate \(U_\textrm{ideal}\) applied on an initial state \(|\psi(0)\rangle\), leading to the final state \(|\psi_\textrm{ideal}\rangle=U_\textrm{ideal}|\psi(0)\rangle\). In practice, the actual final state will be \(|\psi(t)\rangle\), which is given by the solution of the time-dependent Schrödinger’s equation under the actual EDSR Hamiltonian for the initial state \(|\psi(0)\rangle\) (seen above). The fidelity of the quantum-logic gate is then

(12.10)\[F(U_\textrm{ideal},|\psi(0)\rangle, |\psi(t)\rangle) = |\langle \psi(0)|U^\dagger_\mathrm{ideal}|\psi(t)\rangle|^2.\]

This formula for the gate fidelity is implemented in the fidelity method of the Dynamics class.

A simple scenario: a driven two-level system under the rotating-wave approximation

Despite QTCAD being much more general (e.g., by accounting for several single-electron basis states in the system dynamics), it is often useful to focus on simple two-level system dynamics, which can be solved very efficiently, in particular under the rotating-wave approximation.

For sufficiently small gate bias modulations \(\delta\boldsymbol\varphi^\textrm{bias}(t)\), the matrix elements of the perturbation \(\delta \hat V(t)\) to the confinement potential (see Eq. (12.4)) respond linearly to \(\delta\boldsymbol\varphi^\textrm{bias}(t)\). For sinusoidal driving, we may then write Eq. (12.6) as

(12.11)\[\hat H(t) = \hat H_0 + \delta\hat V \cos(\omega t),\]

where \(\omega\) is the drive frequency and \(\delta\hat V\) the operator giving the amplitude of the perturbation due to the drive in the eigenbasis of the Zeeman Hamiltonian.

Projecting into the two-level subspace formed by \(\{|0\rangle,|1\rangle\}\) (here, the integer corresponds to the quantum number \(k\) introduced above, which labels eigenstates of the Zeeman Hamiltonian), we may decompose \(\hat H_0\) and \(\delta\hat V\) over Pauli matrices defined on the logical qubit eigenbasis:

(12.12)\[\tau_x \equiv |0\rangle\langle 1|+|1\rangle\langle 0|, \qquad \tau_y \equiv -i|1\rangle\langle 0| + i |0\rangle\langle 1|, \qquad \tau_z \equiv |1\rangle\langle 1| - |0\rangle\langle 0|.\]

This gives

(12.13)\[\begin{split}\hat H_0 &= \frac{\hbar\omega_q}2\tau_z,\\ \delta\hat V &= \delta V_x\tau_x + \delta V_y\tau_y + \delta V_z\tau_z,\end{split}\]

with \(\omega_q\) being the qubit angular frequency, and where

(12.14)\[\delta V_\alpha = \frac12 \textrm{Tr} \left[\tau_\alpha \delta \hat V\right],\qquad \alpha\;\in\;\{x,y,z\}.\]

The resulting time-dependent Hamiltonian is

(12.15)\[\hat H(t) = \frac{\hbar\omega_q}2\tau_z + \left(\delta V_x\tau_x + \delta V_y\tau_y + \delta V_z\tau_z\right)\cos(\omega t).\]

We then move to the frame that rotates around the \(z\) axis at the drive frequency \(\omega\) using the operator

(12.16)\[U_R(t) = \textrm{exp}\left(-\frac{i}2\omega \tau_z t\right).\]

The solution \(|\psi_R(t)\rangle\equiv U_R^\dagger(t)|\psi(t)\rangle\) of the time-dependent Schrödinger equation in the rotating frame is then obtained from

(12.17)\[\begin{split}H_R(t) |\psi_R(t)\rangle &= i\hbar\frac{\partial}{\partial t}|\psi_R(t)\rangle,\\ H_R(t) &\equiv U_R^\dagger(t) \hat H(t)U_R(t)-\frac{\omega}2\tau_z.\end{split}\]

Substituting Eq. (12.15) into Eq. (12.17) gives

(12.18)\[H_R(t) = \left[ \left(\delta V_x-i\delta V_y\right)\tau_+\textrm e^{i\omega t} + \left(\delta V_x+i\delta V_y\right)\tau_-\textrm e^{-i\omega t} + \delta V_z\tau_z \right]\cos(\omega t) + \frac{\hbar(\omega_q-\omega)}2\tau_z,\]

where \(\tau_+ \equiv |1\rangle\langle 0|\) and \(\tau_- \equiv |0\rangle\langle 1|\). Assuming that the drive frequency \(\omega\) is sufficiently close to the qubit frequency \(\omega_q\), and that the drive amplitude is sufficiently weak, we may decompose the cosine into \(\cos(\omega t)=(\textrm e^{i\omega t}+\textrm e^{-i\omega t})/2\) and neglect high-frequency terms that oscillate as \(\textrm e^{\pm 2i\omega t}\) in Eq. (12.18). This rotating-wave approximation (RWA) results in

(12.19)\[H_R\approx \frac12\delta V_x\tau_x + \frac12\delta V_y\tau_y + \frac{\hbar(\omega_q-\omega)}2\tau_z.\]

Because of its time-independence, Eq. (12.19) is trivial to solve numerically, and may be used to obtain quick estimates of EDSR dynamics. Despite widespread usage of this equation in the literature, the qubit frequency and drive amplitudes are typically obtained empirically from fits to experimental data. In contrast, using QTCAD, it becomes possible to calculate \(\delta V_x\), \(\delta V_y\), and \(\omega_q\) for realistic device geometries before running an EDSR experiment. Indeed, the above Hamiltonian may easily be generated for a realistic device using the H_RF method of the Dynamics class. Examples of usage of this method are given in Electric dipole spin resonance—Noise and Charge noise in quantum dots.

A more realistic scenario: noisy electric-dipole spin resonance

Here, we consider a simple modification of the above scenario in which, in addition to the gate bias modulation used for control, there exists a small classical noise component. More specifically, we consider the Hamiltonian

(12.20)\[\hat H(t) = \hat H_0 + \delta\hat V[1+\beta(t)] \cos(\omega t),\]

where \(\beta(t)\) is a realization of a classical stochastic process modelling the noise. For a zero-mean, Gaussian, stationary, and ergodic process, the noise statistics are fully characterized by the noise power spectral density (PSD)

(12.21)\[S_\beta (\omega) = \int_{-\infty}^\infty d\tau\; \textrm e^{-i\omega \tau}\;\textrm E[\beta(0)\beta(\tau)],\]

where \(\textrm E[\cdot]\) is the classical expectation value of a stochastic process. In Eq. (12.21), \(\textrm E[\beta(0)\beta(\tau)]\) is known as the autocorrelation function of the noise for lag time \(\tau\).

Following the same steps as in A simple scenario: a driven two-level system under the rotating-wave approximation, we expand the operators over Pauli matrices, and invoke the RWA. Here, in addition to assuming sufficiently weak driving at a frequency close to the qubit frequency, we also assume that noise is sufficiently weak and contains negligible spectral content at \(\omega = \omega_q\). In this situation, the final Hamiltonian in the rotating frame is approximately

(12.22)\[H_R(t) \approx \frac12\left[1+\beta(t)\right]\delta V_x\tau_x + \frac12\left[1+\beta(t)\right]\delta V_y\tau_y + \frac{\hbar(\omega_q-\omega)}2\tau_z.\]

The time evolution of the system under this Hamiltonian may then be solved under a certain number of realizations of the stochastic process \(\beta(t)\). Expectation values such as \(\langle \hat O\rangle\) are then replaced by sample means \(\langle \bar{O} \rangle\equiv\sum_m\langle \hat O\rangle/N_\textrm{samples}\) over \(N_\textrm{samples}\) noise-process realizations. This approach yields an increasingly accurate estimate of the expectation value of an operator as \(N_\textrm{samples}\) increases.

Noisy EDSR may be simulated simply using the dynamics method of a Noise object, which implements the time-dependent Hamiltonian given by Eq. (12.22). In addition, this method takes as input an arbitrary noise spectrum. Typical noise spectra (Lorentzian, and \(1/f\)) are readily available from the spectra module.

Finally, examples of usage of the noisy EDSR features are given in Electric dipole spin resonance—Noise and Charge noise in quantum dots.